If you need a shorter hash just truncate a modern hash algorithm down to 160 or 128 bits. Obviously the standard lengths were chosen for a reason, but SHA2-256/160 or SHA2-256/128 are better hash functions than SHA1 or MD5, respectively. Blake2b/160 is even faster than SHA1!
(I suspect this would be a good compromise for git, since so much tooling assumes a 160 bit hash, and yet we don't want to continue using SHA1)